On the Convergence of the Matrix Lambert W Approach to Solution of Systems of Delay Differential Equations

Abstract

Convergence of the matrix Lambert W function method for solving systems of delay differential equations (DDEs) is considered. Recent research shows that convergence problems occur with certain DDEs when using the well-established Q-iteration approach. A complementary, and recently proposed, W-iteration approach is shown to converge even on systems where Q-iteration fails. Furthermore, the role played by the branch numbers k = −∞ … −1, 0, 1, … ∞ of the matrix Lambert W function, Wk, in terms of initializing the iterative solutions, is also discussed and elucidated. Several second-order examples, known to have convergence problems with Q-iteration, are readily solved by W-iteration. Examples of third- and fourth-order DDEs show that W-iteration is also effective on higher-order systems.